Soil Liquid-Phase (Solute) Diffusion
Definition of Diffusion: Mass transport by random atomic motion
Fick’s laws:
Fick’s 1st law:
steady-state conditions
Fick’s 2nd law:
transient conditions
J = solute diffusive flux
D = diffusivity or diffusion coefficient
Subscript l = indicates liquid-phase system
C = solute concentration
x = distance in the x direction
t = time
When is diffusion important?
 During low advective flow and short distances of a system
o Engineering example
 Human-made waste repositories (i.e. dumps) have a compacted soil
lining at the bottom to minimize advective flow. Therefore, if the lining
was successfully constructed and no advective flow occurs the solutes
could then only pass through the liner by diffusion. Important for
environmental contamination.
o Agriculture example
 Nitrogen fertilization in a well aggregated soil. For a mildly wet soil (i.e.
water films along the outside of both soil particles and soil aggregates but
the inter-aggregate spaces are mostly filled with air), immediately after a
nitrogen product is injected into the soil a large concentration gradient of
soil water nitrogen from the outside to the inside of a soil aggregate,
assuming that all the nitrogen was readily dissolved in the soil water or
was already in an aqueous state. So what did we learned from Dr. A.
Fick?.... A diffusive flux can be related to a concentration gradient.
Therefore, the nitrogen will diffuse to the inside of the soil aggregate.
However, if the soil is saturated and you inject your nitrogen product
then the nitrogen, where will the nitrogen diffuse to?... diffuse into the
soil aggregate where you have only a fractional volume and tortuosity or
diffuse across the open water of the inter-aggregate space? The nitrogen
will more readily diffuse across the inter-aggregate space and then into
the soil aggregate; but you have reduced your concentration gradient at
the soil aggregate surface and consequently reduce the diffusive flux and
the amount of nitrogen within the soil aggregate.
So why is this important? It’s important because infiltrating water during
a precipitation event in a well-aggregated soil will have both flowing
water in the inter-aggregate spaces and stagnant water in the intraaggregate spaces. Thus, the more nitrogen product diffused into the soil
aggregate, where the stagnant water is present, and then you will
minimize the loss of nitrogen due to leaching. Important for the plant
nitrogen needs, nitrogen use efficiency, environmental water quality.
Diffusion in soil
 Standard equation for soil liquid-phase diffusivity



D = soil diffusivity
Dw = diffusivity in bulk water
θ = volumetric water content
= tortuosity
However, similar to that of soil-gas phase diffusivity, many mathematical models have
been developed for liquid-phase diffusivity.
is difficult to quantify
Soil particle surfaces can affect diffusivity. Soil particles are dominantly negatively
charged. Therefore cations will accumulate near the soil particle and soil aggregate
surfaces. This accumulation changes the fluid properties of the media your solute is
diffusing through. Hillel gives α, that can be linearly applied to diffusivity, as a factor
accounting for the increased viscosity near the solid surface. Though one may argue
that visocity is not an active mechanism when advection is not present, the fluid
properties near the soil particle surface are different than that farther away and you
would expect the diffusivity to change. Important in soil with large fraction of small
pores.
H2O
H2O
H2O
H2O
Solute
H2O
H2O
H2O
H2O
H2O
H2O
H2O
Solute
H2O
Cation
H2O
Soil Water
H2O
H2O
H2O
H2O Solute
H2O
H2O
H2O
H2O
H2O
Solute
H2O
H2O
H2O
H2O
H2O
Negatively
charged soil
particle
H2O
H2O
H2O
H2O
H2O

Soil diffusivity is a function of θ and .
o As θ decreases, Dl decreases. As increases, Dl decreases
o However is also a function of θ. As θ decreases, increases.
Soil particle
Soil particle
Soil particle
Soil particle
Extras (not responsible for on the Test)
Diffusivity in bulk water
 Stokes-Einstein equation
D = diffusivity
K = Boltzmann constant (a physical constant relating energy at the particle level with
temperature
T = Kelvins
μ = dynamic viscosity of the liquid
r = radius of the solute

Nerst equation
D = diffusivity
R = Molar gas constant
T = Kelvins
= Faraday constant (electric charge carried by one mole of electrons)
= electrical conductivity
= valence of ion
However, ions typically may travel in pairs with differ ions, if present. Anion solutes will
commonly pair with some cation solute and travel together (Ex: NO3- may travel with
free Ca2+ or Na+). Each are attracted to each other due to electrostatic forces. Each are
likely to give a different diffusivity. However they will travel at some average.
+
-
D2
D0
D1